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Simultaneous diophantine approximation for a restricted class of pairs of real numbers

Youssef Lazar

TL;DR

This work advances the Littlewood conjecture by establishing concrete, verifiable conditions under which a pair of badly approximable numbers satisfies the conjecture. It introduces a cubic Diophantine form $F_k(t)$ tied to Dirichlet lattice points near the line $(1,\alpha,\beta)$ and constructs rational lines $L^k_{\alpha,\beta}$ that converge to this line. Through precise asymptotics of the cubic's coefficients and a careful analysis of the roots and the small-value set, the paper ensures the existence of lattice points with arbitrarily small Littlewood form values, under the stated convergent-denominator conditions. The result provides explicit criteria involving the denominators of continued fraction convergents and their least common multiples, enriching the landscape of cases where the Littlewood conjecture is known to hold and offering a path toward broader generalization.

Abstract

We prove that the Littlewood conjecture is satisfied for a restricted class of pairs $(α,β)$ of badly approximable numbers. We use the localization of the roots of a cubic equation with coefficients depending on the diophantine properties for the considered pair $(α,β)$. The estimates of the roots rely on the properties of the denominators of the convergents of the continued fraction expansion of $α$ and $β$.

Simultaneous diophantine approximation for a restricted class of pairs of real numbers

TL;DR

This work advances the Littlewood conjecture by establishing concrete, verifiable conditions under which a pair of badly approximable numbers satisfies the conjecture. It introduces a cubic Diophantine form tied to Dirichlet lattice points near the line and constructs rational lines that converge to this line. Through precise asymptotics of the cubic's coefficients and a careful analysis of the roots and the small-value set, the paper ensures the existence of lattice points with arbitrarily small Littlewood form values, under the stated convergent-denominator conditions. The result provides explicit criteria involving the denominators of continued fraction convergents and their least common multiples, enriching the landscape of cases where the Littlewood conjecture is known to hold and offering a path toward broader generalization.

Abstract

We prove that the Littlewood conjecture is satisfied for a restricted class of pairs of badly approximable numbers. We use the localization of the roots of a cubic equation with coefficients depending on the diophantine properties for the considered pair . The estimates of the roots rely on the properties of the denominators of the convergents of the continued fraction expansion of and .
Paper Structure (12 sections, 8 theorems, 116 equations, 4 figures)

This paper contains 12 sections, 8 theorems, 116 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The main result
  3. Rational lines of approximations and cubic equations
  4. Continued fractions expansion of a real number
  5. Small values of the Littlewood cubic form
  6. Dirichlet lattice points near $\mathcal{D}(\varepsilon)$
  7. Line approximation
  8. The cubic polynomial $F_{k}(t)$ associated to the Dirichlet lattice points $(M_{k})_{k \geqslant 1}$
  9. Estimates of the coefficients of $F_{k}$
  10. Asymptotic shape of the graph $y=F_{k}(t)$
  11. Intervals for the solutions of $F_{k}(t) = \pm \varepsilon$
  12. Proof of Theorem \ref{['main']}

Key Result

Theorem 1.2

Let $(\alpha,\beta)\in \mathbb{B}^{2}$. If one can find a sequence $(\eta_{k})_{k \geq 1}$ of real numbers and a pair of sequences $(n_{k})_{k \geq 1}$, $(m_{k})_{k \geq 1}$ of even positive integers with $0 \leq \eta_{k} < 1/3$ for every $k$, such that the two following conditions hold for $k$ larg Then the pair $(\alpha, \beta)$ satisfies the Littlewood conjecture.

Figures (4)

  • Figure 1: A view of the picture in the plane $z=0$. The red line $(L^{k}_{\alpha, \beta})$ passes through the Dirichlet lattice vector and cuts $\mathcal{D}(\varepsilon)$ at $x(t_{k}(\varepsilon))$. The boundary of $\mathcal{D}(\varepsilon)$ is colored in blue.
  • Figure 2: The shape of $y=F_{k}(t)$
  • Figure 3: The root $x_{k}$ dominates $t_{k}(\alpha)$ and $t_{k}(\beta)$ as soon as $\delta_{k} >4/3$.
  • Figure 4: One the left the worst scenario $|C_{k}(\pm \varepsilon)| <1$ (three intervals) and one the right the best scenario $|C_{k}(\pm \varepsilon)| \geq 1$ (one interval).

Theorems & Definitions (9)

  • Conjecture 1.1: Littlewood
  • Theorem 1.2
  • Corollary 1.3
  • Proposition 2.1
  • Theorem 2.2: Theorem 3.1, VIII,§ 3 lg
  • Proposition 2.3
  • Lemma 3.1
  • Proposition 4.1
  • Lemma 4.2